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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
16 non associative with precedence 45
17 for @{ 'PSubstStar $L $T1 $d $e $T2 }.
19 include "basic_2/substitution/tps.ma".
21 (* PARTIAL UNFOLD ON TERMS **************************************************)
23 definition tpss: nat → nat → lenv → relation term ≝
24 λd,e,L. TC … (tps d e L).
26 interpretation "partial unfold (term)"
27 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
29 (* Basic eliminators ********************************************************)
31 lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
32 (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
33 ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
34 #d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
35 @(TC_star_ind … HT1 IHT1 … HT12) //
38 lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
39 (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
40 ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
41 #d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
42 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
45 (* Basic properties *********************************************************)
47 lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
50 lemma tpss_strap1: ∀L,T1,T,T2,d,e.
51 L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
54 lemma tpss_strap2: ∀L,T1,T,T2,d,e.
55 L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
58 lemma tpss_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
59 ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
62 lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
65 lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
66 ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
67 L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
68 #L #V1 #V2 #d #e #HV12 elim HV12 -V2
69 [ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
71 | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
73 | #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
74 lapply (tpss_lsubr_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
75 lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
79 lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
80 L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 →
81 L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2.
82 #L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
83 [ #V2 #HV12 #HT12 elim HT12 -T2
85 | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
87 | #V #V2 #_ #HV12 #IHV #HT12
88 lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
92 lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 →
93 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
94 L ⊢ T1 ▶* [d2, e2] T2.
95 #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2
97 | #T #T2 #_ #HT12 #IHT
98 lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
102 lemma tpss_weak_top: ∀L,T1,T2,d,e.
103 L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2.
104 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2
106 | #T #T2 #_ #HT12 #IHT
107 lapply (tps_weak_top … HT12) -HT12 /2 width=3/
111 lemma tpss_weak_full: ∀L,T1,T2,d,e.
112 L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2.
113 #L #T1 #T2 #d #e #HT12
114 lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
115 lapply (tpss_weak_top … HT12) //
118 lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
119 ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
120 #K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
123 (* Basic inversion lemmas ***************************************************)
125 (* Note: this can be derived from tpss_inv_atom1 *)
126 lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k.
127 #L #T2 #k #d #e #H @(tpss_ind … H) -T2
129 | #T #T2 #_ #HT2 #IHT destruct
130 >(tps_inv_sort1 … HT2) -HT2 //
134 (* Note: this can be derived from tpss_inv_atom1 *)
135 lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
136 #L #T2 #p #d #e #H @(tpss_ind … H) -T2
138 | #T #T2 #_ #HT2 #IHT destruct
139 >(tps_inv_gref1 … HT2) -HT2 //
143 lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
144 ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 &
145 L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
147 #d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
149 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
150 elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
151 lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
155 lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 →
156 ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 &
158 #d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
160 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
161 elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/
165 lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2.
166 #L #T1 #T2 #d #H @(tpss_ind … H) -T2
168 | #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
172 (* Basic forward lemmas *****************************************************)
174 lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ♯{T1} ≤ ♯{T2}.
175 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
177 lapply (tps_fwd_tw … HT2) -HT2 #HT2
178 @(transitive_le … IHT1) //
181 lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
182 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
183 #L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
185 | #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
186 elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/